We are then in the framework of [1]}, Chapter IV, Theorem 4.2 (compact control set) and we can deduce that \(u \in \mathcal {C}^{1,2}_b([0,T]\times \mathbb {R}^d)\)  is actually a strong solution to the Hamilton-Jacobi equation. In particular, \(Du\)  and \(D^2u\)  are bounded. The second point is precisely [2]} Theorem 4.4.4. We give the argument for the sake of completeness. Since \(u\)  is a strong solution we can linearize the HJB equation and see that \(u\)  satisfies the (linear) PDE :
\(\left\lbrace \begin{array}{ll}\partial _t u + V(t,x).Du + Z(t,x).D^2u = s(x) & \mbox{in } [0,T]\times \mathbb {R}^d \\u(0,x) = g(x) & \mbox{in } \mathbb {R}^d \\\end{array}\right.\) 
